cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A111125 Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 30, 27, 9, 1, 11, 55, 77, 44, 11, 1, 13, 91, 182, 156, 65, 13, 1, 15, 140, 378, 450, 275, 90, 15, 1, 17, 204, 714, 1122, 935, 442, 119, 17, 1, 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1, 21, 385, 2079, 5148, 7007, 5733, 2940, 952, 189, 21, 1
Offset: 0

Views

Author

N. J. A. Sloane, Oct 16 2005

Keywords

Comments

Riordan array ((1+x)/(1-x)^2, x/(1-x)^2). Row sums are A002878. Diagonal sums are A003945. Inverse is A113187. An interesting factorization is (1/(1-x), x/(1-x))(1+2*x, x*(1+x)). - Paul Barry, Oct 17 2005
Central coefficients of rows with odd numbers of term are A052227.
From Wolfdieter Lang, Jun 26 2011: (Start)
T(k,s) appears as T_s(k) in the Knuth reference, p. 285.
This triangle is related to triangle A156308(n,m), appearing in this reference as U_m(n) on p. 285, by T(k,s) - T(k-1,s) = A156308(k,s), k>=s>=1 (identity on p. 286). T(k,s) = A156308(k+1,s+1) - A156308(k,s+1), k>=s>=0 (identity on p. 286).
(End)
A111125 is jointly generated with A208513 as an array of coefficients of polynomials v(n,x): initially, u(1,x)= v(1,x)= 1; for n>1, u(n,x)= u(n-1,x) +x*(x+1)*v(n-1) and v(n,x)= u(n-1,x) +x*v(n-1,x) +1. See the Mathematica section. The columns of A111125 are identical to those of A208508. Here, however, the alternating row sums are periodic (with period 1,2,1,-1,-2,-1). - Clark Kimberling, Feb 28 2012
This triangle T(k,s) (with signs and columns scaled with powers of 5) appears in the expansion of Fibonacci numbers F=A000045 with multiples of odd numbers as indices in terms of odd powers of F-numbers. See the Jennings reference, p. 108, Theorem 1. Quoted as Lemma 3 in the Ozeki reference given in A111418. The formula is: F_{(2*k+1)*n} = Sum_{s=0..k} ( T(k,s)*(-1)^((k+s)*n)*5^s*F_{n}^(2*s+1) ), k >= 0, n >= 0. - Wolfdieter Lang, Aug 24 2012
From Wolfdieter Lang, Oct 18 2012: (Start)
This triangle T(k,s) appears in the formula x^(2*k+1) - x^(-(2*k+1)) = Sum_{s=0..k} ( T(k,s)*(x-x^(-1))^(2*s+1) ), k>=0. Prove the inverse formula (due to the Riordan property this will suffice) with the binomial theorem. Motivated to look into this by the quoted paper of Wang and Zhang, eq. (1.4).
Alternating row sums are A057079.
The Z-sequence of this Riordan array is A217477, and the A-sequence is (-1)^n*A115141(n). For the notion of A- and Z-sequences for Riordan triangles see a W. Lang link under A006232. (End)
The signed triangle ((-1)^(k-s))*T(k,s) gives the coefficients of (x^2)^s of the polynomials C(2*k+1,x)/x, with C the monic integer Chebyshev T-polynomials whose coefficients are given in A127672 (C is there called R). See the odd numbered rows there. This signed triangle is the Riordan array ((1-x)/(1+x)^2, x/(1+x)^2). Proof by comparing the o.g.f. of the row polynomials where x is replaced by x^2 with the odd part of the bisection of the o.g.f. for C(n,x)/x. - Wolfdieter Lang, Oct 23 2012
From Wolfdieter Lang, Oct 04 2013: (Start)
The signed triangle S(k,s) := ((-1)^(k-s))*T(k,s) (see the preceding comment) is used to express in a (4*(k+1))-gon the length ratio s(4*(k+1)) = 2*sin(Pi/4*(k+1)) = 2*cos((2*k+1)*Pi/(4*(k+1))) of a side/radius as a polynomial in rho(4*(k+1)) = 2*cos(Pi/4*(k+1)), the length ratio (smallest diagonal)/side:
s(4*(k+1)) = Sum_{s=0..k} ( S(k,s)*rho(4*(k+1))^(2*s+1) ).
This is to be computed modulo C(4*(k+1), rho(4*(k+1)) = 0, the minimal polynomial (see A187360) in order to obtain s(4*(k+1)) as an integer in the algebraic number field Q(rho(4*(k+1))) of degree delta(4*(k+1)) (see A055034). Thanks go to Seppo Mustonen for asking me to look into the problem of the square of the total length in a regular n-gon, where this formula is used in the even n case. See A127677 for the formula in the (4*k+2)-gon. (End)
From Wolfdieter Lang, Aug 14 2014: (Start)
The row polynomials for the signed triangle (see the Oct 23 2012 comment above), call them todd(k,x) = Sum_{s=0..k} ( (-1)^(k-s)*T(k,s)*x^s ) = S(k, x-2) - S(k-1, x-2), k >= 0, with the Chebyshev S-polynomials (see their coefficient triangle (A049310) and S(-1, x) = 0), satisfy the recurrence todd(k, x) = (-1)^(k-1)*((x-4)/2)*todd(k-1, 4-x) + ((x-2)/2)*todd(k-1, x), k >= 1, todd(0, x) = 1. From the Aug 03 2014 comment on A130777.
This leads to a recurrence for the signed triangle, call it S(k,s) as in the Oct 04 2013 comment: S(k,s) = (1/2)*(1 + (-1)^(k-s))*S(k-1,s-1) + (2*(s+1)*(-1)^(k-s) - 1)*S(k-1,s) + (1/2)*(-1)^(k-s)*Sum_{j=0..k-s-2} ( binomial(j+s+2,s)*4^(j+2)* S(k-1, s+1+j) ) for k >= s >= 1, and S(k,s) = 0 if k < s and S(k,0) = (-1)^k*(2*k+1). Note that the recurrence derived from the Riordan A-sequence A115141 is similar but has simpler coefficients: S(k,s) = sum(A115141(j)*S(k-1,s-1+j), j=0..k-s), k >= s >=1.
(End)
From Tom Copeland, Nov 07 2015: (Start)
Rephrasing notes here: Append an initial column of zeros, except for a 1 at the top, to A111125 here. Then the partial sums of the columns of this modified entry are contained in A208513. Append an initial row of zeros to A208513. Then the difference of consecutive pairs of rows of the modified A208513 generates the modified A111125. Cf. A034807 and A127677.
For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 20 and 21) and Damianou and Evripidou (p. 7).
As suggested by the equations on p. 7 of Damianou and Evripidou, the signed row polynomials of this entry are given by (p(n,x))^2 = (A(2*n+1, x) + 2)/x = (F(2*n+1, (2-x), 1, 0, 0, ... ) + 2)/x = F(2*n+1, -x, 2*x, -3*x, ..., (-1)^n n*x)/x = -F(2*n+1, x, 2*x, 3*x, ..., n*x)/x, where A(n,x) are the polynomials of A127677 and F(n, ...) are the Faber polynomials of A263196. Cf. A127672 and A127677.
(End)
The row polynomials P(k, x) of the signed triangle S(k, s) = ((-1)^(k-s))*T(k, s) are given from the row polynomials R(2*k+1, x) of triangle A127672 by
P(k, x) = R(2*k+1, sqrt(x))/sqrt(x). - Wolfdieter Lang, May 02 2021

Examples

			Triangle T(k,s) begins:
k\s  0    1     2     3     4     5     6    7    8   9 10
0:   1
1:   3    1
2:   5    5     1
3:   7   14     7     1
4:   9   30    27     9     1
5:  11   55    77    44    11     1
6:  13   91   182   156    65    13     1
7:  15  140   378   450   275    90    15    1
8:  17  204   714  1122   935   442   119   17    1
9:  19  285  1254  2508  2717  1729   665  152   19   1
10: 21  385  2079  5148  7007  5733  2940  952  189  21  1
... Extended and reformatted by _Wolfdieter Lang_, Oct 18 2012
Application for Fibonacci numbers F_{(2*k+1)*n}, row k=3:
F_{7*n} = 7*(-1)^(3*n)*F_n + 14*(-1)^(4*n)*5*F_n^3 + 7*(-1)^(5*n)*5^2*F_n^5 + 1*(-1)^(6*n)*5^3*F_n^7, n>=0. - _Wolfdieter Lang_, Aug 24 2012
Example for the  Z- and A-sequence recurrences  of this Riordan triangle: Z = A217477 = [3,-4,12,-40,...]; T(4,0) = 3*7 -4*14 +12*7 -40*1 = 9. A =  [1, 2, -1, 2, -5, 14, ..]; T(5,2) = 1*30 + 2*27 - 1*9 + 2*1= 77. _Wolfdieter Lang_, Oct 18 2012
Example for the (4*(k+1))-gon length ratio s(4*(k+1))(side/radius) as polynomial in the ratio rho(4*(k+1)) ((smallest diagonal)/side): k=0, s(4) = 1*rho(4) = sqrt(2); k=1, s(8) = -3*rho(8) + rho(8)^3 = sqrt(2-sqrt(2)); k=2, s(12) = 5*rho(12) - 5*rho(12)^3 + rho(12)^5, and C(12,x) = x^4 - 4*x^2 + 1, hence rho(12)^5 = 4*rho(12)^3 - rho(12), and s(12) = 4*rho(12) - rho(12)^3 = sqrt(2 - sqrt(3)). - _Wolfdieter Lang_, Oct 04 2013
Example for the recurrence for the signed triangle S(k,s)= ((-1)^(k-s))*T(k,s) (see the Aug 14 2014 comment above):
S(4,1) = 0 + (-2*2 - 1)*S(3,1) - (1/2)*(3*4^2*S(3,2) + 4*4^3*S(3,3)) = - 5*14 - 3*8*(-7) - 128*1 = -30. The recurrence from the Riordan A-sequence A115141 is S(4,1) = -7 -2*14 -(-7) -2*1 = -30. - _Wolfdieter Lang_, Aug 14 2014

Links

Crossrefs

Mirror image of A082985, which see for further references, etc.
Also closely related to triangles in A098599 and A100218.
Cf. A052227, A085478, A208513, A208508, A211957
Cf. A034807, A127672, A127677, A263196.

Programs

  • Magma
    [((2*n+1)/(n+k+1))*Binomial(n+k+1, 2*k+1): k in [0..n], n in [0..12]];  // G. C. Greubel, Feb 01 2022
  • Mathematica
    (* First program *)
u[1, x_]:=1; v[1, x_]:=1; z=16;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208513 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A111125 *) (* Clark Kimberling, Feb 28 2012 *)
(* Second program *)
T[n_, k_]:= ((2*n+1)/(2*k+1))*Binomial[n+k, 2*k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 01 2022 *)
  • Sage
    @CachedFunction
def T(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 3*T(n-1,k) if n==1 else 2*T(n-1,k)
    return T(n-1,k-1) - T(n-2,k) - h
A111125 = lambda n,k: (-1)^(n-k)*T(n,k)
for n in (0..9): [A111125(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
From Peter Bala, Apr 30 2012: (Start)
T(n,k) = binomial(n+k,2*k) + 2*binomial(n+k,2*k+1).
The row generating polynomials P(n,x) are a generalization of the Morgan-Voyce polynomials b(n,x) and B(n,x). They satisfy the recurrence equation P(n,x) = (x+2)*P(n-1,x) - P(n-2,x) for n >= 2, with initial conditions P(0,x) = 1, P(1,x) = x+r+1 and with r = 2. The cases r = 0 and r = 1 give the Morgan-Voyce polynomials A085478 and A078812 respectively. Andre-Jeannin has considered the case of general r.
P(n,x) = U(n+1,1+x/2) + U(n,1+x/2), where U(n,x) denotes the Chebyshev polynomial of the second kind - see A053117. P(n,x) = (2/x)*(T(2*n+2,u)-T(2*n,u)), where u = sqrt((x+4)/4) and T(n,x) denotes the Chebyshev polynomial of the first kind - see A053120. P(n,x) = Product_{k = 1..n} ( x + 4*(sin(k*Pi/(2*n+1)))^2 ). P(n,x) = 1/x*(b(n+1,x) - b(n-1,x)) and P(n,x) = 1/x*{(b(2*n+2,x)+1)/b(n+1,x) - (b(2*n,x)+1)/b(n,x)}, where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. Cf. A211957.
(End)
From Wolfdieter Lang, Oct 18 2012 (Start)
O.g.f. column No. s: ((1+x)/(1-x)^2)*(x/(1-x)^2)^s, s >= 0. (from the Riordan data given in a comment above).
O.g.f. of the row polynomials R(k,x):= Sum_{s=0..k} ( T(k,s)*x^s ), k>=0: (1+z)/(1-(2+x)*z+z^2) (from the Riordan property).
(End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013

Extensions

More terms from Paul Barry, Oct 17 2005

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

Original entry on oeis.org

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522
Offset: 0

Views

Author

Paul Barry, Sep 17 2004

Keywords

Comments

Row sums of A098599.

Links

Crossrefs

Cf. A000032, A000035, A000045, A001350, A008346.
Cf. A020878, A098599, A099925, A181716, A182143.
First differences of A014217 and A062724.

Programs

  • Magma
    [Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 31 2014
  • Mathematica
    Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)
Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)
CoefficientList[ Series[-(1 + 2x)/(-1 + 2x^2 + x^3), {x, 0, 40}], x] (* or *)
LinearRecurrence[{0, 2, 1}, {1, 2, 2}, 40] (* Robert G. Wilson v, Mar 09 2018 *)
  • PARI
    a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014
  • PARI
    Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016
  • SageMath
    [lucas_number2(n,1,-1) -(-1)^n for n in range(51)] # G. C. Greubel, Mar 26 2024

Formula

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).
a(n) = Sum_{k = 0..n} binomial(k, n-k) + binomial(k-1, n-k-1).
a(n) = A020878(n) - 1 = A001350(n) + 1.
a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004
a(n) = A181716(n+1). - Richard R. Forberg, Aug 30 2014
a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015
From Colin Barker, Jun 03 2016: (Start)
a(n) = (-(-1)^n + ((1/2)*(1-sqrt(5)))^n + ((1/2)*(1+sqrt(5)))^n).
a(n) = 2*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = A014217(n) + A000035(n). - Paul Curtz, Jul 27 2023

A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -1, 0, -3, 1, -1, 0, 2, -4, 1, -1, 0, 0, 5, -5, 1, -1, 0, 0, -2, 9, -6, 1, -1, 0, 0, 0, -7, 14, -7, 1, -1, 0, 0, 0, 2, -16, 20, -8, 1, -1, 0, 0, 0, 0, 9, -30, 27, -9, 1, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1
Offset: 0

Views

Author

Paul Barry, Nov 08 2004

Keywords

Examples

			Triangle begins as:
   1;
  -1,  1;
  -1, -2,  1;
  -1,  0, -3,  1;
  -1,  0,  2, -4,  1;
  -1,  0,  0,  5, -5,   1;
  -1,  0,  0, -2,  9,  -6,   1;
  -1,  0,  0,  0, -7,  14,  -7,  1;
  -1,  0,  0,  0,  2, -16,  20, -8,  1;
  -1,  0,  0,  0,  0,   9, -30, 27, -9,  1;

Links

Crossrefs

Cf. A000007, A000027, A077961, A080956, A098601.
Cf. A154955, A157142, A280560, A355021.
Row sums are A100219.
Matrix inverse of A100100.
Apart from signs, same as A098599.
Very similar to triangle A111125.

Programs

  • Magma
    A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;
[A100218(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Mar 28 2024
  • Mathematica
    T[0,0]:= 1; T[1,1]:= 1; T[1,0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1,k] +T[n-1,k-1] -2*T[n-2,k-1] +T[n-3,k-1]]; Table[T[n,k], {n,0,14}, {k,0,n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)
  • SageMath
    def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))
flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Mar 28 2024

Formula

Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 28 2024: (Start)
T(n, n-1) = A000027(n), n >= 1.
T(n, n-2) = -A080956(n-1), n >= 2.
T(2*n, n) = A280560(n).
T(2*n-1, n) = A157142(n-1), n >= 1.
T(2*n+1, n) = -A000007(n) = A154955(n+2).
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
From Peter Bala, Apr 28 2024: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A098601 Expansion of (1+2*x)/((1+x)*(1-x^2-x^3)).

Original entry on oeis.org

1, 1, 0, 3, 0, 4, 2, 5, 5, 8, 9, 14, 16, 24, 29, 41, 52, 71, 92, 124, 162, 217, 285, 380, 501, 666, 880, 1168, 1545, 2049, 2712, 3595, 4760, 6308, 8354, 11069, 14661, 19424, 25729, 34086, 45152, 59816, 79237, 104969, 139052, 184207, 244020, 323260
Offset: 0

Views

Author

Paul Barry, Sep 17 2004

Keywords

Comments

Diagonal sums of A098599.
The signed sequence 1,-1,0,-3,0,-4,... gives the diagonal sums of A100218. - Paul Barry, Nov 09 2004

Links

Crossrefs

Cf. A000931, A077883, A098599, A100218, A135364.

Programs

  • Magma
    I:=[1,1,0,3]; [n le 4 select I[n] else -Self(n-1) +Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..55]]; // G. C. Greubel, Mar 27 2024
  • Mathematica
    CoefficientList[Series[(1+2x)/((1+x)(1-x^2-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{-1,1,2,1},{1,1,0,3},50] (* Harvey P. Dale, Dec 14 2011 *)
  • SageMath
    def A098601(n): return sum( binomial(k, n-2*k) + binomial(k-1, n-2*k-1) for k in range(1+n//2))
[A098601(n) for n in range(56)] # G. C. Greubel, Mar 27 2024

Formula

G.f.: x/((1+x)*(1-x^2-x^3)) + 1/(1-x^2-x^3).
a(n) = Sum_{k=0..floor(n/2)} (binomial(k, n-2*k) + binomial(k-1, n-2*k-1)).
a(n) = -a(n-1) + a(n-2) + 2*a(n-3) + a(n-4).
Inverse binomial transform of A135364. - Paul Curtz, Apr 25 2008

A111126 Triangle read by rows: T(k,s) = binomial(k+s,2s+1)*(2k-1)*(2k+1)/(2s+3), k >= 1, 0 <= s <= k-1.

Original entry on oeis.org

1, 10, 3, 35, 28, 5, 84, 126, 54, 7, 165, 396, 297, 88, 9, 286, 1001, 1144, 572, 130, 11, 455, 2184, 3510, 2600, 975, 180, 13, 680, 4284, 9180, 9350, 5100, 1530, 238, 15, 969, 7752, 21318, 28424, 20995, 9044, 2261, 304, 17, 1330, 13167, 45144, 76076, 72618
Offset: 1

Views

Author

N. J. A. Sloane, Oct 16 2005

Keywords

Examples

			Triangle starts:
1;
10,3;
35,28,5;
84,126,54,7;
165,396,297,88,9;

Links

Crossrefs

Mirror image of A111127. Cf. A111125, A082985, A100218, A098599.

Programs

  • Maple
    T:=(k,s)->binomial(k+s,2*s+1)*(2*k-1)*(2*k+1)/(2*s+3): for k from 1 to 10 do seq(T(k,s),s=0..k-1) od; # yields sequence in triangular form; Emeric Deutsch, Feb 01 2006

Extensions

More terms from Emeric Deutsch, Feb 01 2006

A247492 Triangle read by rows: T(n, k) = binomial(k-1, n-k)*(n+1)/(n+1-k), 0 <= k <= n.

Original entry on oeis.org

1, -1, 2, 1, 0, 3, -1, 0, 2, 4, 1, 0, 0, 5, 5, -1, 0, 0, 2, 9, 6, 1, 0, 0, 0, 7, 14, 7, -1, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 9, 30, 27, 9, -1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, -1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12
Offset: 0

Views

Author

Peter Luschny, Oct 01 2014

Keywords

Examples

			[0]  1;
[1] -1, 2;
[2]  1, 0, 3;
[3] -1, 0, 2, 4;
[4]  1, 0, 0, 5, 5;
[5] -1, 0, 0, 2, 9, 6;
[6]  1, 0, 0, 0, 7, 14, 7;
.
Taylor series: 1 + x*(2*y - 1) + x^2*(3*y^2 + 1) + x^3*(4*y^3 + 2*y^2 - 1) + x^4*(5*y^4 + 5*y^3 + 1) + O(x^5).

Crossrefs

Cf. A001350 (row sums), A098599, A100218.

Programs

  • Maple
    T := (n, k) -> (n+1)*binomial(k-1, n-k)/(n+1-k);
for n from 0 to 11 do seq(T(n,k), k=0..n) od;

Formula

Sum_{k = 0..n} T(n, k) = A001350(n+1).
G.f.: (x^2*y + 1)/((x^4 + 2*x^3 + x^2)*y^2 + (-x^3 - 3*x^2 - 2*x)*y + x + 1). Or: (x^2*y + 1)/((x + 1)*(x*y - 1)*(x^2*y + x*y - 1)). - Vladimir Kruchinin, Oct 23 2021
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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